Calcul du pgcd de deux entiers

𝑎, 𝑏 0< 𝑏 < 𝑎
𝑎, 𝑏 𝑎
𝑏
𝑎, 𝑏
𝑎 𝑏
(𝑎, 𝑏) [2,100]
𝑎 𝑏
(𝐹𝑛)𝑛
𝑎 𝑏
𝑎⩾𝐹𝑛+1 𝑏⩾𝐹𝑛
𝑛
𝑎 𝑏 𝑛
𝑎=𝑏𝑞1+𝑟10< 𝑟1< 𝑏 (1)
𝑏=𝑟1𝑞2+𝑟20< 𝑟2< 𝑟1(2)
𝑟𝑖−2=𝑟𝑖−1𝑞𝑖+𝑟𝑖0< 𝑟𝑖< 𝑟𝑖−1(𝑖)
𝑟𝑛−3=𝑟𝑛−2𝑞𝑛−1+𝑟𝑛−10< 𝑟𝑛−1< 𝑟𝑛−2(𝑛−1)
𝑟𝑛−2=𝑟𝑛−1𝑞𝑛(𝑛)
𝑞𝑛⩾2𝑟𝑛−2⩾2 = 𝐹3
𝑟𝑖−2=𝑟𝑖−1𝑞𝑖+𝑟𝑖⩾𝑟𝑖−1+𝑟𝑖⩾𝐹𝑛−𝑖+1 +𝐹𝑛−𝑖=𝐹𝑛−𝑖+2
𝑎=𝑟−1⩾𝐹𝑛+2 𝑏=𝑟0⩾𝐹𝑛+1 □
(𝐹𝑛)
𝐹𝑛+1 𝐹𝑛+2 𝑛
𝑖 > 0𝐹𝑖>0𝑖 > 1𝐹𝑖+1 > 𝐹𝑖
𝐹𝑛+1 𝐹𝑛

𝐹𝑛+1 =𝐹𝑛+𝐹𝑛−1(1)
𝐹𝑛=𝐹𝑛−1+𝐹𝑛−2(2)
𝐹3= 2𝐹2(𝑛−1)
□
𝑎 𝑏
10 𝑏
𝑛 𝑙(𝑛)
2𝑛
𝑙(𝑛) = 𝐸(log2(𝑎)) 𝑎 > 0
1𝑎= 0
𝑙
𝑎+𝑏 𝑂(𝑙(𝑎) + 𝑙(𝑏))
𝑎𝑏 𝑂(𝑙(𝑎)𝑙(𝑏))
𝐸(𝑎/𝑏)𝑂(𝑙(𝑎)𝑙(𝐸(𝑎/𝑏)))
𝑎 𝑏 𝑂(𝑙(𝑎)𝑙(𝐸(𝑎/𝑏)))
𝑂(𝑙(𝑎)𝑙(𝑏))
𝑛
𝑎=𝑟1𝑏(1)
𝑏=𝑟2𝑟1(2)
𝑟𝑖−2=𝑟𝑖𝑟𝑖−1(𝑖)
𝑟𝑛−2= 0 𝑟𝑛−1(𝑛)
𝑖 𝑂(𝑙(𝑟𝑖−2)𝑙(𝐸(𝑟𝑖−1/𝑟𝑖−2))) = 𝑂(𝑙(𝑟𝑖−2)𝑙(𝑞𝑖+2)))
𝑟𝑖−2=𝑞𝑖𝑟𝑖−1+𝑟𝑖
𝑛−2
𝑖=−1
𝑂(𝑙(𝑟𝑖)𝑙(𝑞𝑖+2))) ⩽𝑂(𝑙(𝑏))(
𝑛−2
𝑖=−1
𝑙(𝑞𝑖+2)) = 𝑂(𝑙(𝑏))𝑙(
𝑛
𝑖=1
𝑞𝑖)) ⩽𝑂(𝑙(𝑏))𝑂(𝑙(𝑎)).
𝑞𝑖𝑟𝑖−1⩽𝑟𝑖−2(𝑛
𝑖=1 𝑞𝑖)⩽𝑟−1/𝑟𝑛−1⩽𝑟−1=𝑎□

𝑎, 𝑏
𝑎
𝑏
(𝑎, 𝑏)𝑏 < 𝑎 [2,100] ×[2,100]
𝑎 𝑏
𝑛
𝑢, 𝑣, 𝑤, 𝑒
𝑢:= 𝑎
𝑣:= 𝑏
𝑒:= 0
𝑢2=0 𝑣2=0
𝑒:= 𝑒+ 1
𝑢:= 𝑢/2
𝑣:= 𝑣/2
(𝑢 > 0)
𝑢2=0
𝑢:= 𝑢/2
𝑣2=0
𝑣:= 𝑣/2
𝑢 < 𝑣
𝑤:= 𝑣 𝑣 := 𝑢 𝑢 := 𝑤
𝑢:= 𝑢−𝑣
2𝑒𝑣
𝑎 𝑏
𝑂(𝑚𝑎𝑥(𝑙(𝑎), 𝑙(𝑏))2)
𝑠 𝑡
𝑠𝑎 +𝑡𝑏 =𝑑 𝑑 𝑎, 𝑏

𝑢, 𝑣, 𝑞, 𝑟, 𝑠, 𝑡, 𝑠1, 𝑡1, 𝑠2, 𝑡2
𝑢:= 𝑎
𝑣:= 𝑏
𝑠1:= 1
𝑡1:= 0
𝑠2:= 0
𝑡2:= 1
𝑟:= 𝑢 𝑣
(𝑟 > 0)
𝑞:= (𝑢−𝑟)/𝑣
𝑢:= 𝑣
𝑣:= 𝑟
𝑠:= 𝑠1𝑡:= 𝑡1𝑠1:= 𝑠2𝑡1:= 𝑡2
𝑠2:= 𝑠−𝑠1𝑞
𝑡2:= 𝑡−𝑡1𝑞
𝑟:= 𝑢 𝑣
(𝑣, 𝑠2, 𝑡2)
𝑛
𝑎 𝑏 𝑛
𝑎 𝑠−1= 1 𝑡−1= 0 (−1)
𝑏 𝑠0= 0 𝑡0= 1 (0)
𝑎=𝑏𝑞1+𝑟10< 𝑟1< 𝑏 𝑠1=𝑠−1−𝑞1𝑠0𝑡1=𝑡−1−𝑞1𝑡0(1)
𝑏=𝑟1𝑞2+𝑟20< 𝑟2< 𝑟1𝑠2=𝑠0−𝑞2𝑠1𝑡2=𝑡0−𝑞2𝑡1(2)
𝑟𝑖−2=𝑟𝑖−1𝑞𝑖+𝑟𝑖0< 𝑟𝑖< 𝑟𝑖−1𝑠𝑖=𝑠𝑖−2−𝑞𝑖𝑠𝑖−1𝑡𝑖=𝑡𝑖−2−𝑞𝑖𝑡𝑖−1(𝑖)
𝑟𝑛−3=𝑟𝑛−2𝑞𝑛−1+𝑟𝑛−10< 𝑟𝑛−1< 𝑟𝑛−2𝑠𝑛−1=⋅⋅⋅ 𝑡𝑛−1=⋅⋅⋅ (𝑛−1)
𝑟𝑛−2=𝑟𝑛−1𝑞𝑛(𝑛)
𝑠𝑖𝑎+𝑡𝑖𝑏=𝑟𝑖𝑠𝑖𝑡𝑖−1−𝑠𝑖−1𝑡𝑖= (−1)𝑖−1𝑝𝑔𝑐𝑑(𝑠𝑖, 𝑡𝑖)=1
∀𝑖
𝑠𝑖𝑠𝑖+1 (∣𝑠𝑖∣)𝑖
𝑡𝑖𝑡𝑖+1 (∣𝑡𝑖∣)𝑖
𝑟𝑖−1∣𝑠𝑖∣⩽𝑏 𝑟𝑖−1∣𝑡𝑖∣⩽𝑎
∣𝑠𝑛−1∣⩽𝑏/2∣𝑡𝑛−1∣⩽𝑎/2
𝑠𝑖𝑡𝑖
𝑞𝑖𝑟𝑖
𝑂
𝑂(𝑚𝑎𝑥(𝑙(𝑎), 𝑙(𝑏))2)

𝑢, 𝑣, 𝑤, 𝑒, 𝑠1, 𝑡1, 𝑠2, 𝑡2, 𝑎2, 𝑏2
𝑢:= 𝑎
𝑣:= 𝑏
𝑒:= 0
𝑠1:= 1
𝑡1:= 0
𝑠2:= 0
𝑡2:= 1
(𝑢 > 0) 𝑢2=0 𝑣2=0
𝑒:= 𝑒+ 1
𝑢:= 𝑢/2
𝑣:= 𝑣/2
𝑎2:= 𝑢
𝑏2:= 𝑣
(𝑢 > 0)
𝑢2=0
𝑢:= 𝑢/2
𝑠12=0 𝑡12=0
𝑠1:= 𝑠1/2
𝑡1:= 𝑡1/2
𝑠1:= (𝑠1+𝑏2)/2
𝑡1:= (𝑡1−𝑎2)/2
𝑣2 = 0
𝑣:= 𝑣/2
𝑠22=0 𝑡22=0
𝑠2:= 𝑠2/2
𝑡2:= 𝑡2/2
𝑠2:= (𝑠2+𝑏2)/2
𝑡2:= (𝑡2−𝑎2)/2
𝑢 < 𝑣 𝑤 := 𝑣 𝑣 := 𝑢 𝑢 := 𝑤
𝑤:= 𝑠1𝑠1:= 𝑠2𝑠2:= 𝑤
𝑤:= 𝑡1𝑡1:= 𝑡2𝑡2:= 𝑤
𝑢:= 𝑢−𝑣
𝑠1:= 𝑠1−𝑠2
𝑡1:= 𝑡1−𝑡2
2𝑒𝑣, 𝑠2, 𝑡2